Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

Robert J.  Martin; Patrizio  Neff

doi:10.3934/jgm.2016010

Article Contents

2016, Volume 8, Issue 3: 323-357. Doi: 10.3934/jgm.2016010

This issue Previous Article The projective symplectic geometry of higher order variational problems: Minimality conditions Next Article An approximation theorem in classical mechanics

Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

Robert J. Martin^1, and
Patrizio Neff^2,

1.
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen
2.
Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen

Received: October 2014

Revised: February 2016

Published: September 2016

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Abstract

We provide an easy approach to the geodesic distance on the general linear group $GL(n)$ for left-invariant Riemannian metrics which are also right-$O(n)$-invariant. The parameterization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.

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Mathematics Subject Classification: 53C22.

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References

[1]	E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group, Journal of Geometry and Physics, 86 (2014), 241-257.doi: 10.1016/j.geomphys.2014.08.009.
[2]	D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition), Princeton reference, Princeton University Press, 2009.doi: 10.1515/9781400833344.
[3]	A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, Journal of Geometric Mechanics, 3 (2011), 197-223.doi: 10.3934/jgm.2011.3.197.
[4]	P. G. Ciarlet, Three-Dimensional Elasticity, Number 1 in Studies in Mathematics and its Applications, Elsevier Science, 1988.
[5]	C. De Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations, Journal of Elasticity, 15 (1985), 225-227.doi: 10.1007/BF00041995.
[6]	M. P. do Carmo, Riemannian Geometry, Birkhäuser Basel, 1992.doi: 10.1007/978-1-4757-2201-7.
[7]	J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen, Springer, 2007.
[8]	S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer, 3 edition, 1990.doi: 10.1007/978-3-642-97242-3.
[9]	H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?, Zeitschrift für Physik, 55 (1929), 145-155. Available from: http://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1929.pdf.
[10]	N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.doi: 10.1137/1.9780898717778.
[11]	J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.), Springer, 1998.doi: 10.1007/978-3-662-22385-7.
[12]	K. Königsberger, Analysis 1, Analysis, Springer, 2004.
[13]	J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor, Linear Algebra and its Applications, 449 (2014), 28-42.doi: 10.1016/j.laa.2014.02.012.
[14]	S. Lee, M. Choi, H. Kim and F. C. Park, Geometric direct search algorithms for image registration, IEEE Transactions on Image Processing, 16 (2007), 2215-2224.doi: 10.1109/TIP.2007.901809.
[15]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17. Springer, 1999.doi: 10.1007/978-0-387-21792-5.
[16]	R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$, in preparation, 2016.
[17]	A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002, 61-90.doi: 10.1007/0-387-21791-6_2.
[18]	M. Moakher, Means and averaging in the group of rotations, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 1-16.doi: 10.1137/S0895479801383877.
[19]	P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations, Technical report, Technische Universität Darmstadt, 2008. available at https://www.uni-due.de/ hm0014/Download_files/cism_convexity08.pdf.
[20]	P. Neff, B. Eidel and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics, Archive for Rational Mechanics and Analysis, 222 (2016), 507-572.doi: 10.1007/s00205-016-1007-x.
[21]	P. Neff, B. Eidel, F. Osterbrink and R. Martin, A Riemannian approach to strain measures in nonlinear elasticity, Comptes Rendus Mécanique, 342 (2014), 254-257.doi: 10.1016/j.crme.2013.12.005.
[22]	P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral and frobenius norms, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1132-1154.doi: 10.1137/130909949.
[23]	C. H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2011.doi: 10.1093/acprof:oso/9780199605880.001.0001.
[24]	B. Vandereycken, P.-A. Absil and S. Vandewalle, A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank, IMA Journal of Numerical Analysis, 33 (2013), 481-514.doi: 10.1093/imanum/drs006.